31. Let X₁, . . ., X_n be independent and identically distributed random variables having distribution function F and density

f. The quantity M = [X₍₁₎ + X_(n)]/2, defined to be the average of the smallest and largest value, is called the midrange. Show that its distribution function is

$$F_M(m) = n \int_0^m [F(2m - x) - F(x)]^{n-1} f(x) dx$$